Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 119, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
MULTIPLICITY OF CRITICAL POINTS FOR THE
FRACTIONAL ALLEN-CAHN ENERGY
DAYANA PAGLIARDINI
Abstract. In this article we study the fractional analogue of the Allen-Cahn
energy in bounded domains. We show that it admits a number of critical
points which approaches infinity as the perturbation parameter tends to zero.
1. Introduction
The problems involving fractional operators attracted great attention during
the previous years. Indeed these problems appear in areas such as optimization,
finance, crystal dislocation, minimal surfaces, water waves, fractional diffusion; see
for example [8, 6, 3, 4, 7, 19, 18]). In particular, from a probabilistic point of view,
the fractional Laplacian is the infinitesimal generator of a Lévy process, see e.g. [2].
In this article we present some existence and multiplicity results for critical points
of functionals of the form
Z Z
Z
|u(x) − u(y)|2
1
F (u) =
W (u) dx, if s ∈ (0, 1/2),
(1.1)
dx dy + 2s
n+2s
Ω Ω |x − y|
Ω
Z Z
Z
1
|u(x) − u(y)|2
1
F (u) =
dx
dy
+
W (u) dx, if s = 1/2,
| log | Ω Ω |x − y|n+1
| log | Ω
(1.2)
Z
2s−1 Z Z
2
|u(x) − u(y)|
1
F (u) =
dx dy +
W (u) dx, if s ∈ (1/2, 1), (1.3)
n+2s
2
Ω
Ω Ω |x − y|
where Ω is a smooth bounded domain of Rn , u ∈ H s (Ω; R), W ∈ C 2 (R; R+ ) is the
well known double well potential (see Section 2), and ∈ R+ .
F is the fractional energy of the Allen-Cahn equation. It is the fractional counterpart of the functionals studied by Modica-Mortola in [14, 15] where they proved
the Γ-convergence of the energy to De Giorgi’s perimeter. In the same way, functionals (1.1), (1.2), (1.3) have been also considered by Valdinoci-Savin in [17], where
it is discussed their Γ-convergence.
Moreover, as proved in [13] for the functional
Z
[|Du|2 + −1 (u2 − 1)2 ] dx,
Ω
2010 Mathematics Subject Classification. 35R11, 58J37.
Key words and phrases. Allen-Cahn energy; fractional PDE; critical point; genus.
c 2016 Texas State University.
Submitted March 4, 2016. Published May 13, 2016.
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2
D. PAGLIARDINI
EJDE-2016/119
we expect that the solutions have interesting geometric properties related to the
interface minimality.
Some authors investigated multiplicity results of nontrivial solution for
2s (−∆)s u + u = h(u)
in Ω
u>0
u=0
(1.4)
on ∂Ω
n
where Ω is a bounded domain in R , n > 2s, and h(u) has a subcritical growth (see
[12]), or for
2s (−∆)s u + V (z)u = f (u) in Rn , n > 2s
u ∈ H s (Rn )
(1.5)
n
u(z) > 0 z ∈ R
where the potential V : Rn → R and the nonlinearity f : R → R satisfy suitable
assumptions (see [11]).
Then Cabré and Sire in [5] studied the equation
(−∆)s u + G0 (u) = 0
in Rn
where G denotes the potential associated to a nonlinearity f , and they proved
existence, uniqueness and qualitative properties of solutions.
Indeed, Passaseo in [16] studied the analogue of our functional, with the classical
Laplacian instead of the fractional one, i.e.,
Z
Z
1
G(u) dx
(1.6)
f (u) = |Du|2 dx +
Ω
Ω
where Ω is a bounded domain of Rn , u ∈ H 1,2 (Ω; R), G ∈ C 2 (R; R+ ) is a nonnegative function having exactly two zeros, α and β, and is a positive parameter: he
proved that the number of critical points for f goes to ∞ as → 0.
Passaseo was motivated by De Giorgi’s idea, contained in [9], i.e. if u → u0 in
L1 (Ω) as → 0 and lim→0 f (u ) < ∞, then the function U (t), defined as steepest
descent curves for f starting from u , converge to a curve U0 (t) in L1 (Ω) such that
U0 (t) is a function with values in {α, β} for every t ≥ 0 and the interface between
the sets Et = {x ∈ Ω : U0 (t)(x) = α} and Ω \ Et moves by mean curvature. As a
consequence the critical points u of f which satisfy
lim inf f (u ) < +∞
→∞
(1.7)
converge in L1 (Ω) to a function u0 taking values in {α, β}. De Giorgi considered
also the problem of existence and multiplicity for nontrivial critical points of f
with the property (1.7), and Passaseo’s critical points verify this property and
lim inf f (u ) > 0,
→∞
so he can say that u0 is nontrivial.
In this article we want to extend Passaseo’s results by replacing the function G
in (1.6) with the double well potential W , and Passaseo’s functional f with its
fractional counterpart.
The article is organized as follows: in Section 2 we give some preliminaries
definitions and results. In Section 3, we define suitable functions and sets, then
most of the work is dedicated to prove nonlocal estimates needful to obtain the
bound from above of F , (see Lemma 3.5), and the (PS)-condition, Lemma 3.7. In
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CRITICAL POINTS OF THE ALLEN-CAHN ENERGY
3
fact in particular for the first of these results, we had to split the domain in two
types of regions and estimate F in the three possible interactions.
Finally, after recalling a technical result, Lemma 3.6, we can apply a classical
Krasnoselsii’s genus tool to show the existence and multiplicity results for solutions.
Hence, knowing that minimizers of F Γ-converge to minimizers of the area
functional, we hope that also min-max solutions can pass to the limit as → 0 in
a suitable sense, producing critical points of positive index for local, if s ∈ [1/2, 1),
or nonlocal, if s ∈ (0, 1/2), area functional.
2. Notation and preliminary results
In this section we introduce the framework that we will be used throughout this
article.
Let Ω be a bounded domain of Rn , denote by |Ω| its Lebesgue measure and
consider W the double well potential, that is an even function such that
W : R → [0, +∞)
W ∈ C 2 (R; R+ ) W (±1) = 0
W > 0 in (−1, 1) W 0 (±1) = 0 W 00 (±1) > 0.
(2.1)
Now we fix the fractional exponent s ∈ (0, 1). For any p ∈ [1, +∞), we define
o
n
|u(x) − u(y)|
∈ Lp (Ω × Ω) ;
W s,p (Ω) := u ∈ Lp (Ω) :
s+n/p
|x − y|
i.e. an intermediary Banach space between Lp (Ω) and W 1,p (Ω), endowed with the
natural norm
Z Z
Z
1/p
|u(x) − u(y)|p
kukW s,p (Ω) :=
|u|p dx +
dx
dy
.
n+sp
Ω
Ω Ω |x − y|
If p = 2 we define W s,2 (Ω) = H s (Ω) and it is a Hilbert space. Now let S 0 (Rn ) be
the set of all temperated distributions, that is the topological dual of S (Rn ). As
usual, for any ϕ ∈ S (Rn ), we denote by
Z
1
e−iξ·x ϕ(x) dx
F ϕ(ξ) =
n/2
Rn
(2π)
the Fourier transform of ϕ and we recall that one can extend F from S (Rn ) to
S 0 (Rn ). At this point we can define, for any u ∈ S (Rn ) and s ∈ (0, 1), the
fractional Laplacian operator as
Z
u(x) − u(y)
s
(−∆) u(x) = C(n, s)P.V.
dy.
n+2s
n
R |x − y|
Here P.V. stands for the Cauchy principal value and C(n, s) is a normalizing constant (see [10] for more details). It is easy to prove that this definition is equivalent
to the following two:
Z
1
u(x + y) + u(x − y) − 2u(x)
(−∆)s u(x) = − C(n, s)
dy ∀x ∈ Rn ,
n+2s
2
|y|
n
R
and
(−∆)s u(x) = F −1 (|ξ|2s (F u)) ∀ξ ∈ Rn .
Now we recall some embedding’s results for the fractional spaces:
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D. PAGLIARDINI
EJDE-2016/119
Proposition 2.1 ([10]). Let p ∈ [1, +∞) and 0 < s ≤ s0 ≤ 1. Let Ω be an open
0
set of Rn and u : Ω → R be a measurable function. Then W s ,p (Ω) is continuously
0
embedded in W s,p (Ω), denoted by W s ,p (Ω) ,→ W s,p (Ω), and the following inequality
holds
kukW s,p (Ω) ≤ CkukW s0 ,p (Ω)
for some suitable positive constant C = C(n, s, p) ≥ 1.
Moreover, if also Ω is an open set of Rn of class C 0,1 with bounded boundary,
then W 1,p (Ω) ,→ W s,p (Ω) and we have
kukW s,p (Ω) ≤ CkukW 1,p (Ω)
for some suitable positive constant C = C(n, s, p) ≥ 1.
Definition 2.2 ([10]). For any s ∈ (0, 1) and any p ∈ [1, +∞), we say that an open
set Ω ⊆ Rn is an extension domain for W s,p if there exists a positive constant C =
C(n, p, s, Ω) such that: for every function u ∈ W s,p (Ω) there exists ũ ∈ W s,p (Rn )
with ũ(x) = u(x) for all x ∈ Ω and kũkW s,p (Rn ) ≤ CkukW s,p (Ω) .
Theorem 2.3 ([10]). Let s ∈ (0, 1) and p ∈ [1, +∞) be such that sp < n. Let
q ∈ [1, p∗ ), where p∗ = p∗ (n, s) = np/(n − sp) is the so-called “fractional critical
exponent”. Let Ω ⊆ Rn be a bounded extension domain for W s,p and I be a
bounded subset of Lp (Ω). Suppose that
Z Z
|f (x) − f (y)|p
sup
dx dy < ∞.
n+sp
f ∈I Ω Ω |x − y|
Then I is pre-compact in Lq (Ω).
We recall also the notion of Krasnoselskii’s genus, useful in the sequel.
Definition 2.4 ([1]). Let H be a Hilbert space and E be a closed subset of H \{0},
symmetric with respect to 0 (i.e. E = −E).
We call genus of E in H, indicated with genH (E), the least integer m such that
there exists φ ∈ C(H; R) such that φ is odd and φ(x) 6= 0 for all x ∈ E.
We set genH (E) = +∞ if there are no integer with the above property and
genH (∅) = 0.
It is well known that genH (S k ) = k + 1 if S k is a k-dimensional sphere of H with
centre in zero.
Finally we recall a well known result:
Theorem 2.5 ([1]). Let H be a Hilbert space and f : H → R be an even C 2 functional satisfying the following Palais-Smale condition: given a sequence (ui )i
in H such that the sequence (f (ui ))i is bounded and f 0 (ui ) → 0, (ui )i is relatively
compact in H.
Set f c = {u ∈ H : f (u) ≤ c} for all c ∈ R. Then, for all c1 , c2 ∈ R, such that
c1 ≤ c2 < f (0), we have
genH (f c2 ) ≤ genH (f c1 ) + #{(−ui , ui ) : c1 ≤ f (ui ) ≤ c2 , f 0 (ui ) = 0},
(2.2)
where, if A is a set, we indicate with #A the cardinality of A.
For the rest of this article, we consider H s (Ω) as Hilbert space and we shall write
simply gen(E) instead of genH s (Ω) (E); then we refer to the Palais-Smale condition
with the symbol (P S)-condition.
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CRITICAL POINTS OF THE ALLEN-CAHN ENERGY
5
3. Multiplicity of critical points
Let us state the fundamental result of the paper.
Theorem 3.1. Let Ω be a smooth bounded domain of Rn and W be a function
satisfying (2.1). Then there exist two sequences of positive numbers (k )k , (ck )k
such that for every ∈ (0, k ), the functional F has at least k pairs
(−u1, , u1, ), . . . , (−uk, , uk, )
of critical points, all of them different from the constant pair (−1, 1) satisfying
−1 ≤ ui, (x) ≤ 1
∀x ∈ Ω, ∀ ∈ (0, k ), i = 1, . . . k;
F (ui, ) ≤ ck
∀ ∈ (0, k ), i = 1, . . . , k.
Moreover, for all ∈ (0, k ) and all i = 1, . . . , k we have
Z
s
F (u) ≥ min F (u) : u ∈ H (Ω), −1 ≤ u(x) ≤ 1 for x ∈ Ω,
u dx = 0 .
(3.1)
Ω
Remark 3.2. The constant function u ≡ 0 is obviously a critical point for the
functional F for every > 0 but it is not included among the ones given by
Theorem 3.1. Instead if s ∈ (1/2, 1), but for the other cases it is similar,
1
W (0)|Ω| → +∞ as → 0.
Moreover, since inf{W (t) : W 0 (t) = 0, −1 < t < 1} > 0, one can say that the
critical points given by Theorem 3.1 are not constant functions. In fact, if u = c
is a constant critical point for F (distinct from −1 and 1), it must be W 0 (c ) = 0
and −1 < c < 1; therefore
F (0) =
W (c ) ≥ inf{W (t) : W 0 (t) = 0, −1 < t < 1} > 0
(3.2)
and so, for example by considering the functional related to s ∈ (1/2, 1), but the
other cases are similar,
1
W (c )|Ω| → +∞ as → 0,
in contradiction with F (c ) ≤ ck for all ∈ (0, k ).
Notice that for all > 0,
Z
s
min F (u) : u ∈ H (Ω), −1 ≤ u(x) ≤ 1 ∀x ∈ Ω,
u dx = 0 > 0
F (c ) =
(3.3)
(3.4)
Ω
if we assume, without loss of generality, that Ω is a connected domain.
Let ū be a minimizing function; if we assume F (ū) = 0, then
Z Z
|ū(x) − ū(y)|2
dx dy ≡ 0
n+2s
Ω Ω |x − y|
and W (ū) ≡ 0. Therefore we must have ū ≡ 0 in contradiction with W (0) > 0.
Definition 3.3. Let k be a fixed positive integer; for every λ = (λ(0) , . . . , λ(k) ) ∈
Rk+1 define the function ϕλ : R → R by
ϕλ (t) =
k
X
m=0
λ(m) cos(mt).
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D. PAGLIARDINI
EJDE-2016/119
For every λ ∈ Rk+1 with |λ|Rk+1 = 1 and > 0, let L (ϕλ ) : R → R be the function
defined by
Z
1 t+ ϕλ (τ )
L (ϕλ )(t) =
dτ ;
2 t− |ϕλ (τ )|
notice that L (ϕλ ) is well defined because ϕλ has only isolated zeros ∀λ ∈ Rk+1
with |λ|Rk+1 = 1.
For x = (x1 , · · · , xn ) ∈ Ω ⊂ Rn we consider the projection onto the first component, P1 (x) = x1 , and the set
Sk = {L (ϕλ ) ◦ P1 : λ ∈ Rk+1 , |λ|Rk+1 = 1}.
Lemma 3.4. Let us fix a, b ∈ R with a < b and set
χ(ϕλ ) = #{t ∈ [a, b] : ϕλ (t) = 0}
for λ ∈ R
k+1
with |λ|Rk+1 = 1. Then for every k ∈ N we have
sup{χ(ϕλ ) : λ ∈ Rk+1 , |λ|Rk+1 = 1} < +∞.
For a proof of the above lemma, see [16].
Lemma 3.5. Let Ω be a bounded domain of Rn and W be a function satisfying
(2.1). Then, for every k ∈ N there exists a positive constant ck such that
max F (f ) ≤ ck
Sk
Proof. Let uλ, = L (ϕλ ) ◦ P1 ∈
∀f ∈ Sk .
(3.5)
and set
a = inf P1 (Ω),
b = sup P1 (Ω),
Zλ = {t ∈ [a, b] : ϕλ (t) = 0},
Zλ, = {t ∈ R : dist(t, Zλ ) < }.
Note that
(i) If P1 (x) ∈
/ Zλ, , then |uλ, (x)| = 1 and Duλ, (x) = 0, while
(ii) if P1 (x) ∈ Zλ, , then |uλ, (x)| ≤ 1 and |Duλ, (x)| ≤ 1 .
Since Ω is bounded, we can suppose it is included in a cube Q of side large enough.
C
the complement to Zλ, , then we have to distinguish
We will denote with Yλ, = Zλ,
three cases:
(a) if x ∈ Yλ, and y ∈ Yλ, ;
(b) if x ∈ Zλ, and y ∈ Yλ, ;
(c) if x ∈ Zλ, and y ∈ Zλ, .
We set k = max{χ(ϕλ ) : λ ∈ Rk+1 , |λ|Rk+1 = 1}, then
Zλ, =
k
X
i
Zλ,
i=1
Now we call Žλ, =
P1−1 (Zλ, )
and Yλ, =
k
X
i
Yλ,
.
i=1
P1−1 (Yλ, )
∩ Ω, Y̌λ, =
∩ Ω and we observe that
Z
W (uλ, ) dx = 0,
(3.6)
Y̌λ,
while if we set ρ = sup{|x| : x ∈ Ω}, M = max{W (t) : |t| ≤ 1} and denote by ωn−1
the (n − 1)-dimensional measure of the unit sphere of Rn−1 , it results
Z
W (uλ, ) dx ≤ M |Žλ, | ≤ 2M ωn−1 ρn−1 .
(3.7)
Žλ,
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CRITICAL POINTS OF THE ALLEN-CAHN ENERGY
7
R R |u (x)−uλ, (y)|2
At this point it remains to analyze Ω Ω λ,|x−y|n+2s
dxdy. We split it in
three cases:
Case (a). We have
Z
Z
Z
k Z
X
|uλ, (x) − uλ, (y)|2
|uλ, (x) − uλ, (y)|2
dxdy
=
dx dy
n+2s
j
i
|x − y|
|x − y|n+2s
Y̌λ, Y̌λ,
i,j=1 Y̌λ, Y̌λ,
i6=j
(3.8)
We denote Q− = Q ∩ P1−1 ({x1 < 0}), Q+ = Q ∩ P1−1 ({y1 > 2}) and we split Q−
in N strips of width , with N of order 1/, so we obtain
Z
k Z
X
|uλ, (x) − uλ, (y)|2
dx dy
j
i
|x − y|n+2s
i,j=1 Y̌λ, Y̌λ,
i6=j
Z
Z
4
dx dy
n+2s
Q− Q+ |x − y|
Z −2 Z +∞
2
≤ 4N k
r−2s−1 dr dx1
≤ k2
2
= N k2
s
(3.9)
−
−2x1
−2
Z
(−2x1 )−2s dx1 .
−
Now we distinguish two cases:
(j) if s 6= 1/2, we have
Z −2
2
21−2s N k 2 1−2s 1−2s
2
Nk
(−2x1 )−2s dx1 =
(2
− 1);
s
s(1 − 2s)
−
(3.10)
(jj) while, if s = 1/2,
2
N k2
s
Z
−2
−
(−2x1 )−2s dx1 =
21−2s
N k 2 log 2.
s
i
i
, so
⊆ Q \ Žλ,
Case (b). We note that Y̌λ,
Z
Z
|uλ, (x) − uλ, (y)|2
dxdy
|x − y|n+2s
Žλ, Y̌λ,
Z
k Z
X
|uλ, (x) − uλ, (y)|2
≤
dxdy
i
i
|x − y|n+2s
i=1 Žλ, Q\Žλ,
Z
k
X
min{1/2 |x − y|2 , 4}
≤ 2ωn−1 ρn−1 sup
dy
i
|x − y|n+2s
i
i=1 x∈Žλ, Q\Žλ,
Z +∞
Z 2 1
n−1
1−2s
−1−2s
≤ 2kωn−1 ρ
r
dr
+
4r
dr
2
0
2
2 r2−2s 2
r−2s +∞ =k
+ 8
ωn−1 ρn−1
2 − 2s 0
−2s 2
22−2s
22−2s = k1−2s
+
ωn−1 ρn−1 .
1−s
s
(3.11)
(3.12)
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D. PAGLIARDINI
EJDE-2016/119
Case (c). It results
|uλ, (x) − uλ, (y)|2
dxdy
|x − y|n+2s
Žλ, Žλ,
Z
k Z
X
|uλ, (x) − uλ, (y)|2
=
dxdy
i
i
|x − y|n+2s
i=1 Žλ, Žλ,
Z
k Z
X
|uλ, (x) − uλ, (y)|2
+
dxdy.
j
i
|x − y|n+2s
i,j=1 Žλ, Žλ,
Z
Z
(3.13)
i6=j
Concerning the first term of the right-hand side, we have
k Z
X
|uλ, (x) − uλ, (y)|2
dxdy
i
i
|x − y|n+2s
i=1 Žλ, Žλ,
Z 2
k
1 X i
22−2s 1−2s
≤ 2
|Žλ, |
r1−2s dr ≤ kωn−1 ρn−1
.
i=1
1−s
0
Z
(3.14)
The other term is estimated as in Case (b).
So we can obtain the estimates for the functionals F . In fact, by (3.9), (3.10),
(3.11), (3.12) and (3.14), if s ∈ (0, 1/2) we have
Z
|uλ, (x) − uλ, (y)|2
1
dx
dy
+
W (uλ, ) dx
|x − y|n+2s
2s Ω
Ω Ω
22−2s
22−2s 1−2s ≤ 2kωn−1 ρn−1
1−2s +
1−s
s
kM
22−2s
kωn−1 ρn−1 + 2s 2ωn−1 ρn−1
+ 1−2s
1−s
1−2s
2
2
N k 1−2s 1−2s
(2
− 1)
+
s(1 − 2s)
23−2s
23−2s
22−2s
≤k
+
+
+ 2M ωn−1 ρn−1
1−s
s
1−s
21−2s N k 2 1−2s
+
(2
− 1);
s(1 − 2s)
Z Z
F (uλ, ) =
(3.15)
if s = 1/2 we have
Z Z
Z
|uλ, (x) − uλ, (y)|2
1
1
W (uλ, ) dx
F (uλ , ) =
dx dy +
| log | Ω Ω
|x − y|n+1
| log | Ω
20k
2kM 1
≤
+
ωn−1 ρn−1 +
N k 2 log 2
| log | | log |
s| log |
1
≤ k(20 + 2M )ωn−1 ρn−1 + N k 2 log 2;
s
(3.16)
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CRITICAL POINTS OF THE ALLEN-CAHN ENERGY
9
and, if s ∈ (1/2, 1) we get
Z Z
Z
2s−1
1
|uλ, (x) − uλ, (y)|2
dx
dy
+
W (uλ, ) dx
2
|x − y|n+2s
Ω
Ω Ω
22−2s
22−2s
21−2s
+
+
+ 2M ωn−1 ρn−1
≤k
1−s
s
1−s
2−2s N k 2 1−2s
+
(2
− 1).
s(1 − 2s)
F (uλ , ) =
(3.17)
Now we show a technical lemma, that we will used for proving our main result.
Lemma 3.6. For every > 0 and k ∈ N the set Sk verifies the following properties:
(a)
(b)
(c)
(d)
Sk is a compact subset of H s (Ω);
Sk = −Sk ;
for all k ∈ N there exists ¯k > 0 such that 0 ∈
/ Sk ∀ ∈ (0, ¯k );
k
for all k ∈ N and ∀ > 0 such that 0 ∈
/ S , it results gen(Sk ) ≥ k + 1.
Proof. The points (b), (c) and (d) are proved in [16]. For (a) we use [16, Lemma 2.8]
and the continuous embedding of H 1 (Ω) in H s (Ω) for all s ∈ (0, 1), see Proposition
2.1.
Before proving the main theorem of this work, we point out a useful property of
F .
Lemma 3.7. The functionals (1.1), (1.2), (1.3) satisfy the (PS)-condition.
Proof. We will prove the lemma for s ∈ (1/2, 1) being the other cases analogue. If
W is quadratic, in particular there exist α, β > 0 such that
W (u) ≥ αu + β
∀u ∈ R.
(3.18)
Since (F (un ))n is bounded, (3.18) implies that kun kH s (Ω) is bounded, hence un *
2n
u in H s (Ω), un → u in Lq , ∀q ∈ [1, 2∗ = n−2s
) from Theorem 2.3, therefore un → u
a.e. x ∈ Ω.
We claim that u is a critical point of F . In fact for all v ∈ H s (Ω),
Z Z
u(x) − u(y)
0
2s−1
F (u)(v) = (v(x) − v(y)) dx dy
|x − y|n+2s
Z Ω Ω
1
+
W 0 (u)v dx
Ω
(3.19)
Z Z
un (x) − un (y)
= 2s−1 lim
(v(x)
−
v(y))
dx
dy
n→∞ Ω Ω |x − y|n+2s
Z
1
+ lim
W 0 (un )v dx
n→∞ Ω
since un * u in H s (Ω), un → u in L2 (Ω) and, by hypothesis, F0 (un ) → 0.
10
D. PAGLIARDINI
EJDE-2016/119
This implies that F0 (un )(un − u) + F0 (u)(un − u) → 0, but on the other hand
F0 (un )(un − u) + F0 (u)(un − u)
Z Z
un (x) − un (y)
2s−1
(un (x) − u(x) − un (y) + u(y)) dx dy
=
n+2s
Ω Ω |x − y|
Z Z
u(x) − u(y)
− 2s−1
(un (x) − u(x) − un (y) + u(y)) dx dy
n+2s
Ω Ω |x − y|
Z
1
+
[W 0 (un ) − W 0 (u)(un − u)] dx,
Ω
(3.20)
and the second term on the right hand side appraoches 0. In particular we obtain
Z Z
Z Z
|u(x) − u(y)|2
|un (x) − un (y)|2
dx
dy
→
dx dy.
n+2s
|x − y|n+2s
Ω Ω |x − y|
Ω Ω
Hence kun kH s (Ω) → kukH s (Ω) and since un * u in H s (Ω), we have the result.
We are now able to prove our main result.
Proof of Theorem 3.1. As usual we prove the theorem only for s ∈ (1/2, 1). Consider W ∈ C 2 (R; R+ ) another even function, which satisfies the following properties:
W = W ∀t ∈ [−1, 1];
0
tW (t) > 0 for |t| > 1,
and the asymptotic behaviour guaranteeing that
Z Z
Z
2s−1
|u(x) − u(y)|2
1
F (u) =
W (u) dx
dx dy +
n+2s
2
Ω
Ω Ω |x − y|
is a C 2 -functional satisfying the (PS)-condition.
We prove now that for every critical point u ∈ H s (Ω) which is a critical point
for the functional F , it results |u(x)| ≤ 1 for all x ∈ Ω, and so u is a critical point
for the functional F too: indeed we have that for all v ∈ H s (Ω),
Z Z
Z
u(x) − u(y)
1
0
2s−1
W (u)v dx = 0.
(v(x) − v(y))dxdy +
n+2s
|x
−
y|
Ω Ω
Ω
In particular, if we set û = max{min{u, 1}, −1}, by choosing v = u − û,
Z Z
u(x) − u(y)
2s−1
(u(x) − û(x) − u(y) + û(y)) dx dy
n+2s
Ω Ω |x − y|
Z
1
0
+
W (u)(u − û) dx = 0
Ω
with
u(x) − u(y)
(u(x) − û(x) − u(y) + û(y)) dx dy
n+2s
Ω Ω |x − y|
Z Z
|u(x) − u(y)|2
=
dx dy ≥ 0
n+2s
Ω Ω |x − y|
Z Z
(3.21)
and
Z
0
W (u)(u − û) dx > 0
if u − û 6≡ 0
in Ω
Ω
0
since tW (t) > 0 for |t| > 1. It follows that u = û, i.e., |u(x)| ≤ 1 for almost every
x ∈ Ω.
EJDE-2016/119
CRITICAL POINTS OF THE ALLEN-CAHN ENERGY
11
Let k > 0 be such that k < c1k W (0)|Ω|, where ck is the constant introduced
in Lemma 3.5. Then, for every ∈ (0, k ) we can apply Theorem 2.5 to the functional F with c1 < 0 and c2 = ck , because F (0) = 1 W (0)|Ω| > ck for all
∈ (0, k ). In this way we can prove that for every ∈ (0, k ), F has at least
(k + 1) pairs (−u0, , u0, ), . . . , (−uk, , uk, ) of critical points with F (ui, ) ≤ ck for
c1
ck
all i = 0, 1, . . . , k. In fact gen(F ) = gen(∅) = 0, while gen(F ) ≥ gen(Sk ) ≥ k + 1
c
k
because Sk ⊆ F ⊆ H s (Ω) \ {0}.
Note that these (k + 1) pairs of critical points include also the one implied by
the minimizers ±1; so we can assume that (−u0, , u0, ) = (−1, +1).
On the contrary, the other solutions are not minimizers for the functional F if
Ω is a connected domain. Indeed it results
F (ui, ) > 0
∀ ∈ (0, k ) and i = 0, 1, . . . , k
because if F (ui, ) = F (ui, ) = 0, then we should have
Z Z
|ui, (x) − ui, (y)|2
dx dy = 0 and W (ui, ) ≡ 0
|x − y|n+2s
Ω Ω
in Ω
and so ui, should be a constant function with value +1 or −1.
Moreover let us remark that for all ∈ (0, k ) and i = 1, . . . k we have
Z
s
F (ui, ) ≥ min F (u) : u ∈ H (Ω),
u dx = 0 .
(3.22)
Ω
In fact, we assume that
s
Z
min F (u) : u ∈ H (Ω),
u dx = 0 > 0,
Ω
otherwise (3.22) would be obvious. Then, for every c1 > 0 such that
Z
s
c1 < min F (u) : u ∈ H (Ω),
u dx = 0 ,
Ω
c1
gen(F )
= 1 because below c1 the mean is non zero and we
we would have clearly
can use it as odd function into R1 in the genus definition, see Definition 2.4; thus, if
(3.22) were false, the solutions would belong to a set of genus one, in contradiction
with their construction in Theorem 2.5.
Now, to prove (3.1), let us replace the function W appearing in the definition
j
of functional F by a sequence of functions (W j )j and denote by (F )j the corresponding sequence of new functionals. Assume moreover that the functions W j
satisfy the same properties as W for all j ∈ N and that
lim W j (t) = +∞ for |t| > 1.
(3.23)
j→∞
j
Then property (3.22) holds for the higher critical values of the functional F for all
j ∈ N and so (3.1) follows for j large enough, taking into account that
Z
j
lim min F (u) : u ∈ H s (Ω),
u dx = 0
j→∞
Ω
Z
s
= min F (u) : u ∈ H (Ω), |u(x)| ≤ 1 ∀x ∈ Ω,
u dx = 0
Ω
because of (3.23).
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D. PAGLIARDINI
EJDE-2016/119
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Dayana Pagliardini
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
E-mail address: dayana.pagliardini@sns.it